Letter
pubs.acs.org/JPCL
Dynamics of Charge-Transfer Processes with Time-Dependent
Density Functional Theory
J. I. Fuks,† P. Elliott,‡ A. Rubio,†,§ and N. T. Maitra*,⊥
†
Nano-Bio Spectroscopy Group, Departamento Fı ́sica de Materiales, Universidad del Paı ́s Vasco, Centro de Fı ́sica de Materiales
CSIC-UPV/EHU-MPC and DIPC, Av. Tolosa 72, E-20018 San Sebastián, Spain
‡
Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle (Saale), Germany
§
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany
⊥
Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park
Avenue, New York, New York 10065, United States
S Supporting Information
*
ABSTRACT: We show that whenever an electron transfers
between closed-shell molecular fragments, the exact correlation
potential of time-dependent density functional theory develops a
step and peak structure in the bonding region. This structure has a
density dependence that is nonlocal both in space and in time that
even the exact adiabatic ground-state exchange−correlation
functional fails to capture it. For charge-transfer between openshell fragments, an initial step and peak vanish as the charge-transfer state is reached. The inability of usual approximations to
develop these structures leads to inaccurate charge-transfer dynamics. This is illustrated by the complete lack of Rabi oscillations
in the dipole moment under conditions of resonant charge transfer for an exactly solvable model system. The results transcend
the model and are applicable to more realistic molecular complexes.
SECTION: Molecular Structure, Quantum Chemistry, and General Theory
C
space. This is the case in photovoltaic materials (organic,
inorganic, and hybrids), photocatalysis, biomolecules in
solvents, reactions at the interface between different materials,
and nanoscale conductance devices (see, e.g., refs 13−18 and
references therein). These processes are clearly nonlinear and
require a nonperturbative time-resolved study of electron
dynamics rather than a simple calculation of their excitation
spectrum. TDDFT is increasingly used, often within an
Ehrenfest or surface-hopping scheme, to handle coupled
electron−ion motion.2,13,14,16−18 In the TDDFT scheme, a
one-body time-dependent Kohn−Sham (KS) potential is used
to evolve a set of noninteracting KS electrons, reproducing the
exact one-body density of the true interacting system, from
which all properties of the interacting system may be exactly
extracted. In practice, approximations are required for the xc
potential, υxc[n;Ψ0,Φ0](r,t), a functional of the one-body
density n, the initial interacting state Ψ0, and the initial KS
state Φ0. Almost all calculations today use an adiabatic
approximation that inserts the instantaneous density into a
g.s.
ground-state xc approximation, υadia
xc [n;Ψ0,Φ0](r,t) = υxc [n(t)](r,t), neglecting the dependence of υxc on the past history and
initial states.2 Further, the exact υxc has, in general, a nonlocal
dependence on space.19
harge-transfer (CT) dynamics play a critical role in many
processes of interest in physics, chemistry, and biochemistry, from photochemistry to photosynthesis, solar cell
design, and biological functionality. The quantum mechanical
treatment of such systems calls for methods that can treat
electron correlations and dynamics efficiently for relatively large
systems. Time-dependent density functional theory
(TDDFT)1,2 is the leading candidate today and has achieved
an unprecedented balance between accuracy and efficiency in
calculations of electronic spectra.2,3 CT excitation energies over
medium to large distances are, however, notoriously underestimated by the usual exchange−correlation (xc) functionals,
and recent years have witnessed intense development of many
methods to treat it.4−7 There is recent optimism for obtaining
accurate CT excitations between closed-shell fragments,5,6 but
no functional approximation developed so far works for CT
between open-shell fragments.8−10 Here, standard approximations predict even an unphysical ground state with fractional
occupation in the dissociation limit. For open-shell fragments,
the exact ground-state correlation potential has step and peak
structures,11,12 while the exact xc kernel has strong frequency
dependence and diverges as a function of the fragment
separation; lack of these features in the xc approximation is
responsible for their poor predictions.
In contrast to linear response phenomena, the description of
photoinduced processes generally requires a complete electron
transfer from one state to another or from different regions of
© XXXX American Chemical Society
Received: December 17, 2012
Accepted: February 8, 2013
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The Journal of Physical Chemistry Letters
Letter
A critical question is: Are the available functionals suitable for
modeling the CT processes mentioned earlier? In this Letter,
we show that when an electron transfers at long range from a
ground to an excited CT state, a time-dependent step and peak
are generic and essential features of the exact xc potential.
When the donor and acceptor are both closed shells, the initial
xc potential has no step nor peak, but a step and peak structure
in the bond midpoint region builds up over time. Although in
the initial stages of the CT dynamics the usual approximations
may perform well, they are increasingly worse as time evolves,
leading to completely wrong long-time dynamics. On the other
hand, when the donor and acceptor are both open-shell species,
an initial step and peak structure wanes. Thus, these timedependent steps and peaks that are difficult to capture in
functional approximations play a significant role in CT even
between neutral closed-shell f ragments, unlike in the calculation of
excitation energies. Further, we show that although an adiabatic
approximation to the xc potential may yield a step structure, the
step will, at best, be of the wrong size. Accompanying the step
and peak associated with CT there is also a dynamical step,21
that depends on how the CT is achieved. The exact υxc thus has
a complicated nonlocal space and time dependence that
adiabatic functionals fail to capture, with severe consequences
for time-resolved CT. Although our results are demonstrated
for two electrons, we expect that they can be generalized to real
molecular systems as many cases of CT dominantly involve two
valence electrons. The other electrons act as a general buffer
that introduces some additional dynamical screening that can
change the net size of the step and peak but not their presence.
To illustrate the mechanism of CT processes and the
relevance of spatial and time nonlocality, we use a “two-electron
molecule” in one-dimension. The Hamiltonian is (atomic units
are used throughout)
H(x1 , x 2 , t ) = −
Figure 1. Density (black solid), υS (red long-dashed), υC (blue
dashed), and υext (pink dotted) for the ground state (left) and for the
CT state (right) in our model molecule of closed-shell fragments at
separation R = 7 au.
x
2)1/2ei∫ dx′ u(x′,t), where u(x,t) = j(x,t)/n(x,t) is the local
“velocity”. Inverting the KS equation yields the exact KS
potential as
υS(x , t ) =
−
−Z
(x + R /2)2 + 1
−
(3)
(4)
where υH(x,t) = ∫ dx′ n(x′,t)υee(x − x′) is the Hartree potential
and the external field is given by υext(x,t) = υmol(x) + , (t)x.
Further, for this case, υC = υxc − υX may easily be isolated
because υX = −υH/2.
Before discussing the dynamics, we first consider the final CT
state and focus on CT between closed-shell fragments. Let us
assume that we have complete transfer of an electron at some
time T into the excited state Ψ* (for example, applying a
tailored laser pulse), and the system then stays in this state for
all times t > T. The density, n(t > T) = n*, is then static in the
excited state and nodeless, and the current and velocity u(x,t)
are zero. It follows that the exact υxc(t > T) is static and that the
exact KS potential is given by first two terms of eq 3 only.
In Figure 1, we show the density and the exact KS and
correlation potentials for the ground and CT states for R = 7
au. A clear step and peak structure has developed in the
correlation potential in the region of low density between the
ions in the CT state. There is no such structure in the initial
potential of the ground state. As the separation increases, the
step in υC saturates to a size
(1)
U0
2
cosh (x − R /2)
∂tu(x′, t ) dx′
υxc(x , t ) = υS(x , t ) − υext(x , t ) − υH(x , t )
where υee(y) = 1/(y2 + 1)1/2 is the “soft-Coulomb” electron−
electron interaction22−28 and , (t) = A cos(ωt) is an applied
electric field. The molecule is modeled by
υmol(x) =
x
The xc potential is then
1 ∂2
1 ∂2
−
+ υmol(x1) + υmol(x 2)
2
2 ∂x1
2 ∂x 22
+ υee(x1 − x 2) + ,(t ) ·(x1 + x 2)
∫
∂ 2xn(x , t )
(∂ n(x , t ))2
u 2 (x , t )
− x2
−
2
4n(x , t )
8n (x , t )
(2)
Asymptotically, the soft-Coulomb potential (donor) on the left
decays as −Z/x, similar to a true atomic potential in 3D, while
the cosh-squared (acceptor) on the right is short-ranged,
decaying exponentially away from the “atom”. The acceptor
potential mimics a closed-shell atom without core electrons.
We model CT between two closed-shell fragments by choosing
Z = 2 and U0 = 1, such that, at large separations R, the ground
state has two electrons on the donor and zero on the acceptor,
while the first singlet excited state, Ψ*, is a CT excited state
with one electron in each well (see Figure 1). Choosing Z =
2,U0 = 1.5 places one electron in each well in the ground state,
with a CT excited state having both electrons in the acceptor
well; such a system would model CT between two open-shell
fragments.
If we start the KS simulation in a doubly occupied singlet
state, the KS evolution retains this form for all later times,
Φ(x1,x2,t) = ϕ(x1,t)ϕ(x2,t). Requiring the exact density to be
reproduced at all times leads to ϕ(x,t) = (n(x,t)/
Δ = |IDND− 1 − I ANA + 1|
INDD−1
(5)
IN=1
D
where
=
is the ionization energy of the donor
containing one electron, INA A+1 = IN=1
is that of the one-electron
A
acceptor ion, and the result is written for a general ND(NA)electron donor(acceptor). Equation 5 can be shown by
considering the asymptotics of the donor and acceptor orbitals,
adapting the argument made for the case of the ground state of
a molecule made of open-shell fragments.11,12 Here, instead, we
have a step in the potential of a CT excited state of a molecule
made of closed-shell fragments.
Somewhat of the reverse picture occurs for the case of CT
between open shells; the initial ground-state correlation
potential contains a step and peak, as shown earlier in refs
11, 12, and 29−31, that disappears as the CT state is reached.
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The Journal of Physical Chemistry Letters
Letter
the adiabatically exact xc potential is not the same as the exact
xc potential; the initial-state dependence in the exact functional
reflects a nonlocal time-dependence that persists forever. In the
infinite separation limit, we expect Ψ* and Ψgs
CT to be very
similar, both having a Heitler−London form with one electron
in each well, but the fact that Ψ* is an excited state is encoded
in the nodal structure of its wave function. The correlation
potential is extremely sensitive to this tiny difference in the two
interacting wave functions, which accounts for the different step
size in Figure 2.
The magnitude of the step in υadia−ex
in the infinite separation
xc
limit can be derived by examining the terms in eq 6. In this
limit, locally around each well, υadia
ext must equal the atomic
potential, up to a spatial constant, in order for Ψgs
CT[n*] to
satisfy Schrödinger’s equation there. It cannot simply be the
sum of the atomic potentials because the ground-state Ψ0 of
that potential (eq 2) places two electrons in the donor well. For
adia
Ψgs
CT to be the ground state, υext has a step in the region of
negligible density that pushes up the donor well relative to the
acceptor well; the size of this step, C, is the lowest such that
energetically, it is favorable to place one electron on each well,
as Ψgs
CT[n*] does. Therefore
In either case, the step is a signature of the strong correlation
due to the delocalization of the KS orbital.
The step requires a spatially nonlocal density dependence in
the correlation functional, as in the ground-state case.11,12,29−31
The inability of usual ground-state approximate functionals to
capture this step results in them incorrectly predicting
fractionally charged species. In the present case, we have an
excited state of the interacting system, where the KS orbital
corresponding to the excited-state density n* shown in Figure 1
is in fact a ground-state orbital, ϕ(x) = (n*(x)/2)1/2 because n*
has no nodes. Given the static ground-state nature of the orbital
and KS potentials after time T, does the adiabatic
approximation become exact?
To answer this, we examine the adiabatically-exact xc
potential for t > T, υadia−ex
[n*], that is, evaluating the exact
xc
ground-state xc functional on the instantaneous CT density.
This is (see refs 35 and 36)
adia
υxcadia−ex [n] = υSadia[n] − υext
[n] − υH[n]
υadia
ext [n]
(6)
(υadia
S [n])
where
is the external (exact ground-state KS)
potential for two interacting electrons in a ground state of this
density (υadia
S [n] corresponds to first two terms of eq 3). Figure
2 shows vadia−ex
[n*] for two separations R = 7 and 10 au (see
C
E Dgs, N = 1 + EAgs, N = 1 + C < E Dgs, N = 2 + 2C
(9)
Egs,N
D(A)
where
is the ground-state energy of the N-electron
donor(acceptor). This leads to
C ≥ E Dgs, N = 1 + EAgs, N = 1 − E Dgs, N = 2 = IDND − I ANA + 1
(10)
where in the last line, we have generalized the result to a donor
(acceptor) with ND (NA) electrons.
Now that we have the step in υadia
ext [n*], we use eq 6 to
exact
quantify the step in υadia−ex
[n*]. Because υadia
here, eq 5
xc
S = υS
adia−ex
tells us that the step in υC
is
Figure 2. The exact υC (dashed blue line) and the adiabatically-exact
(red solid line) for R = 7 (left) and 10 au(right). Note that the
υadia−ex
C
potential eventually rolls back down to zero far enough away from the
system. In the infinite separation limit, Δ (Δadia) is given by eq 5 (eq
11).
Δadia = |IDND− 1 − ADND− 1|
which is equal to the derivative discontinuity of the (ND − 1)electron donor. (As before, the entire step is contained in the
N=1
correlation potential.) For our system, IN=1
D = 1.483 au, AD =
N=1
0.755 au, and IA = 0.5 au; thus, in the infinite separation limit,
we get a step of 0.983 (0.729) au in the exact υC(υadia
C ). The
numerical results verify this analysis; the steps shown in Figure
2 for separation R = 7 (10) au have values of 0.61 (0.76) au in
the exact υC and 0.42 (0.55) au in υadia
C . For larger separations,
the steps tend toward the asymptotic values predicted by the
analysis above.
In the above analysis, the adiabatically-exact potential was
evaluated on the exact density, as is commonly done when
assessing functionals,35 rather than on that obtained from a selfconsistent adiabatic propagation. The latter would likely lead to
an erroneous density at time T, but the analysis shows that even
with the exact density at time T, the wrong step size means that
subsequent propagation using the adiabatically exact potential
will yield the wrong dynamics.
Having studied how the xc potential looks for the final CT
state, we now study how the potential evolves in time to reach
such a state. To simplify the analysis, we exploit Rabi physics to
reduce this problem to a two-state system. This approach is
justified for a weak resonant driving field and verified
numerically by comparing the results with the exact timedependent wave function found using octopus.32−34 The
interacting wave function may be written as |Ψ(t)⟩ = ag(t)|Ψgs⟩
+ ae(t)|Ψ*⟩, where
the Supporting Information for numerical methods). Evidently,
the adiabatic approximation does yield a step, but of the wrong
size.
To understand this, first consider the functional dependence
of the exact xc potential. We may write 37
gs
υxc[n](t > T ) = υxc[n*, Ψ*, ΦCT
](t > T )
(7)
where, on the left, the dependence is on the entire history of
the density, n(0 < t < T), and initial-state dependence is not
needed because at t = 0, we start from the ground state.2,37 On
the right, time T is considered as the “initial” time, and the
functional depends on just the static density n* after this time
but, crucially, on the interacting state and KS states at time T.
The former is the CT excited state Ψ*, while the latter is the
doubly occupied orbital, Φ(x1,x2,T) = [n*(x1)n*(x2)]1/2/2 
Φgs
CT, a ground-state wave function, as discussed above.
On the other hand, the adiabatic approximation
gs
gs
υxcadia[n*] ≡ υxcadia[n*, ΨCT
, ΦCT
]
(11)
(8)
differs from the exact xc potential eq 7 in its dependence on the
time-T interacting state; here Ψgs
CT is the ground-state wave
function of an interacting system with density n*, not the true
excited-state wave function. Therefore, eqs 7 and 8 show that
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The Journal of Physical Chemistry Letters
⎛ ag (t )⎞ ⎛ Eg − dgg ,(t ) −deg ,(t ) ⎞⎛ ag (t )⎞
⎟⎜
⎟=⎜
⎟
i∂t⎜⎜
⎟ ⎜
⎟
⎟⎜
⎝ ae(t ) ⎠ ⎝ −deg ,(t ) Ee − dee ,(t )⎠⎝ ae(t ) ⎠
Letter
(12)
with deg = dge = 0.231, dgg = 7, and dee = 0 for our system. The
electric field is resonant with the first excitation, , (t) = 0.006
cos(0.112t).
Figure 3 displays the correlation potential at snapshots in
time over a half-Rabi period TR/2.38 The step, accompanied by
Figure 4. Absolute values of dipole moments |d(t)| for the CT
between closed-shell fragments at separation R = 7 au for exact (solid
black line), adiabatic exact echange (AEXX) (dashed red line), and the
self-interaction-corrected adiabatic local density approximation (SICALDA) (dotted blue line). The calculations were performed in the
presence of a resonant field of frequency ω = 0.112 au and amplitude
A = 0.00667 au.
Note that the CT step recedes asymptotically far from the
molecule,11,12 while the dynamical step persists.21 The relation
of these structures to the derivative discontinuities of the xc
kernel for CT excitations7 will also be investigated in the future.
In modeling real systems, the vibronic coupling introduces a
mixture of excited states that are not, in principle, fully
populated. Still, our findings apply because for an ensemble of
states, CT steps and dynamical steps appear that account for
the population of each excited state contributing to the wave
packet. Note that the step responsible for the CT appears as
soon as the state starts to be populated. Recent work has shown
that TDDFT describes the CT process in an organic
photovoltaic;42 our findings may explain the observed
incomplete CT of the electron to the fullerene.43 The step
feature is a fundamental one for describing processes where
electron−hole splitting is key. Our work highlights an essential
new feature that must be considered in the development of
nonadiabatic functionals able to capture dynamical electrontransfer processes in molecular nanostructures and materials for
energy applications.
Figure 3. (Upper panel) The correlation potential (dotted blue line)
and density (solid black) shown at snapshots of time indicated. (Lower
panel) VC snapshots over an optical cycle centered at around TR/8.
a peak develops over time as the excited CT state is reached; at
TR/2, the correlation potential agrees with the static prediction
earlier (Figure 2, left). Notice that making a time-dependent
constant shift does not affect the dynamics; it just adds a timedependent overall phase. During the second half of the Rabi
cycle, the step gradually disappears. A closer inspection
indicates that superimposed to this smoothly developing step
is an oscillatory step structure, whose dynamics is more on the
time scale of the optical field (lower panel). This faster,
nonadiabatic, nonlocal dynamical step appears generically in
electron dynamics, as shown in ref 21. To distinguish between
the two steps, we refer to the more gradually developing step
due to CT as the CT step.
The impact that the development of the CT step has on
dynamics is significant. The same adiabatic approximations that
for local resonant excitations showed faster but still Rabi-like
oscillations39 fail dramatically to capture any Rabi-like
oscillations between the ground and CT state. This is illustrated
by the dipole moments, d(t) = ⟨ψ(t)|x̂1 + x̂2|ψ(t)⟩, in Figure 4.
The approximate correlation functionals lack the nonlocal
spatial dependence necessary to develop the CT step.40
Given the ubiquity of CT dynamics in topical applications of
TDDFT, it is critical to develop approximations with spatially
nonlocal and nonadiabatic dependence. None of the available
functionals today capture the peak and step structure that
develops in the exact υC as the charge transfers, and they lead to
drastically incorrect dynamics, as illustrated in Figure 4. Even an
exact adiabatic approximation will be incorrect; a step and peak
feature are captured but of the wrong size. The performance of
a self-consistent propagation in such a potential is left for a
future investigation, as is the role of the peak that accompanies
the step. Superimposed on the development of the CT step are
the generic dynamical step and peak features of ref 21; these
features depend on the details of how the CT is induced, for
example, oscillating on the time scale of a resonant optical field.
■
ASSOCIATED CONTENT
S Supporting Information
*
All numerical aspects of the calculations are contained in the
Supporting Information. This material is available free of charge
via the Internet at http://pubs.acs.org.
■
■
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS
We gratefully acknowledge financial support from the U.S.
Department of Energy, Office of Basic Energy Sciences,
Division of Chemical Sciences, Geosciences and Biosciences
under Award DE-SC0008623 (N.T.M.) and a grant of
computer time from the CUNY High Performance Computing
Center under NSF Grant CNS-0855217. The European
Research Council (ERC-2010-AdG-267374), Spanish:
FIS2011-65702-C02-01), Grupos Consolidados (IT-319-07),
and EC Projects CRONOS (280879-2) and CNS-0958379 are
738
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The Journal of Physical Chemistry Letters
Letter
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(43) Rozzi, C. A.; Rubio, A. Private communication.
gratefully acknowledged. J.I.F. acknowledges support from an
FPI fellowship (FIS2007-65702-C02-01).
■
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